Coxeter groups are abstract algebraic structures that arise in various areas of mathematics, including geometry, group theory, and combinatorics. They are defined by a particular type of presentation that involves reflections across hyperplanes in Euclidean space, but they can also be studied in a more abstract way.
Bruhat order is a partial order on the elements of a Coxeter group, particularly related to the symmetric group and general linear groups. It provides a way to compare the "sizes" or "positions" of elements based on their factorizations into simple reflections.
In mathematics, particularly in the study of reflection groups and Coxeter groups, a **Coxeter element** is a specific type of element that is associated with the generating reflections of a Coxeter group. More formally, a Coxeter group is defined by a set of generators that satisfy certain relations, typically corresponding to reflections across hyperplanes in a geometric space. A Coxeter element is typically constructed by taking a set of generators of the Coxeter group and forming their product in a specific order.
A Coxeter group is a special type of group that can be defined geometrically using reflections in Euclidean spaces. These groups are named after H.S.M. Coxeter, who studied their properties and relationships to various geometrical structures. ### Basic Definition: A Coxeter group is defined by a set of generators subjected to specific relations. These relations are based on the angles between the reflections corresponding to the generators.
A Coxeter–Dynkin diagram is a graphical representation used to describe finite and infinite reflection groups, which are important in various areas of mathematics, including geometry, algebra, and theoretical physics. These diagrams are named after mathematicians Harold Scott MacDonald Coxeter and Jacques Dynkin. ### Key Features of Coxeter–Dynkin Diagrams: 1. **Vertices**: Each vertex of the diagram represents a simple root of a root system associated with a given reflection group.
In the context of Coxeter groups, the **longest element** refers to a particular element of the group that can be identified based on its maximal length with respect to the generating set specified by the Coxeter diagram. A **Coxeter group** is defined by a set of generators and relations that can be represented by a diagram (called a Coxeter diagram) where each generator corresponds to a vertex.
A reflection group is a mathematical concept in the field of group theory, specifically in the study of symmetry. It is a type of group that consists of reflections across hyperplanes in a given vector space. Reflection groups can be thought of as the symmetries of geometric objects that can be achieved through reflections. ### Definitions and Properties: 1. **Reflections**: A reflection in a vector space is a linear transformation that flips points across a hyperplane.

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